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We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmannequations for the CMB photon distribution function are studied in three Lorentzframes. We clarify the relations of the SZ effects among the different Lorentzframes. We derive analytic expressions for the photon redistribution functions.These formulas are applicable to the nonthermal electron distributions as wellas the standard thermal distribution. We show that the Fokker-Planck expansionof the Boltzmann equation can be expanded by the power series of the diffusionoperator of the original Kompaneets equation.

Analytical studies on the Sunyaev Zeldovich effect in the cluster of galaxies for three Lorentz frames